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Bulletin of the Korean Mathematical Society

ON QUANTUM CODES FROM CYCLIC CODES OVER A CLASS OF NONCHAIN RINGS

Mustafa Sarı, İrfan Şiap

Published Nov 30, 2016• 25 citations
Publisher

Abstract

In this paper, we extend the results given in [3] to a nonchain ring $R_p={\mathbb{F}}_p+v{\mathbb{F}}_p+{\cdots}+v^{p-1}{\mathbb{F}}_p$ , where $v^p=v$ and p is a prime. We determine the structure of the cyclic codes of arbitrary length over the ring $R_p$ and study the structure of their duals. We classify cyclic codes containing their duals over $R_p$ by giving necessary and sufficient conditions. Further, by taking advantage of the Gray map ${\pi}$ defined in [4], we give the parameters of the quantum codes of length pn over ${\mathbb{F}}_p$ which are obtained from cyclic codes over $R_p$ . Finally, we illustrate the results by giving some examples.

DOI
10.4134/bkms.b150544

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Let Fq\mathbb{F}_qFq​ be a finite field of q=pmq=p^mq=pm elements where ppp is a prime and mmm is a positive integer. This paper considers (γ,Δ)(γ,Δ)(γ,Δ)-cyclic codes over a class of finite non-chain commutative rings Rq,s=Fq[v1,v2,…,vs]/⟨vi−vi2,vivj=vjvi=0⟩\mathscr{R}_{q,s}=\mathbb{F}_q[v_1,v_2,\dots,v_s]/\langle v_i-v_i^2,v_iv_j=v_jv_i=0\rangleRq,s​=Fq​[v1​,v2​,…,vs​]/⟨vi​−vi2​,vi​vj​=vj​vi​=0⟩ where γγγ is an automorphism of Rq,s\mathscr{R}_{q,s}Rq,s​, ΔΔΔ is a γγγ-derivation of Rq,s\mathscr{R}_{q,s}Rq,s​ and 1≤i≠j≤s1\leq i\neq j\leq s1≤i=j≤s for a positive integer sss. Here, we show that a (γ,Δ)(γ,Δ)(γ,Δ)-cyclic code of length nnn over Rq,s\mathscr{R}_{q,s}Rq,s​ is the direct sum of (θ,ℑ)(θ,\Im)(θ,ℑ)-cyclic codes of length nnn over Fq\mathbb{F}_qFq​, where θθθ is an automorphism of Fq\mathbb{F}_qFq​ and ℑ\Imℑ is a θθθ-derivation of Fq\mathbb{F}_qFq​. Further, necessary and sufficient conditions for both (γ,Δ)(γ,Δ)(γ,Δ)-cyclic and (θ,ℑ)(θ,\Im)(θ,ℑ)-cyclic codes to contain their Euclidean duals are established. Then, we obtain many quantum codes by applying the dual containing criterion on the Gray images of these codes. These codes have better parameters than those available in the literature. Finally, the encoding and error-correction procedures for our proposed quantum codes are discussed.
QUANTUM CODES FROM CYCLIC CODES OVER Fq+uFq+vFq\mathbb{F}_{q}+u \mathbb{F}_{q}+v \mathbb{F}_{q}Fq​+uFq​+vFq​
In this paper, we investigate the method of constructing quantum codes from cyclic codes over the ring R=Fq+uFq+vFqR= \mathbb{F}_{q}+u \mathbb{F}_{q}+v \mathbb{F}_{q}R=Fq​+uFq​+vFq​ (u2=u, v2=v, uv=vu=0). By means of the generator polynomials and some decomposition of cyclic codes and their dual over R, we give a necessary and sufficient condition of the cyclic codes over R to be construct the quantum codes. Also as an application, we obtain new non binary quantum codes from those classes of cyclic codes.
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Let R 2 denotes the ring F 2 + μF 2 + υ 2 + μυ F 2 + wF 2 + μwF 2 + υwF 2 + μυwF 2 . In this study, we construct quantum codes from cyclic codes over the ring R 2 , for arbitrary length n, with the restrictions μ 2 = 0, υ 2 = 0, w 2 = 0, μυ = υμ, μw = wμ, υw = wυ and μ (υw) = (μυ) w. Also, we give a necessary and sufficient condition for cyclic codes over R 2 that contains its dual. As a final point, we obtain the parameters of quantum error-correcting codes from cyclic codes over R 2 and we give an example of quantum error-correcting codes form cyclic codes over R 2 .
For any odd prime p, and a divisor ℓ of p, we consider Ie to be the set of all divisors of p - 1, which are less than or equal to ℓ. In this letter, we construct quantum codes from cyclic codes over Fpand FpSℓ, where Sℓ= Πi∈IℓRi, for Ri= (Fp[u])/(〈ui+1-u〉). For that, first we construct linear codes and a Gray map over Rℓ. Using this construction, we study cyclic codes over Rℓ, and then extend that over FpSℓ. We also give a Gray map over FpSℓ. Then, using necessary and sufficient condition of dual containing property for cyclic codes, we construct quantum MDS codes. It is observed that the quantum codes constructed are new in the literature.